Models for Social, Ecological, and Biological Systems: Narrowing the Gap Between Theory and Applications

这项研究旨在开发数学工具，以了解和预测各种重要的和明显不同的生物现象和社会学过程的行为，如疾病的传播、生态变化或犯罪的分布和预防。这些不同现象的一些显著特征可以用基于反应-对流-扩散方程的数学模型来描述，这些数学模型已被广泛用于研究生物学的几个领域中的基本和普遍现象，最近在社会科学中也是如此。虽然这些模型是现实的简化版本，但它们的数学分析有助于理解许多重要的现象。同时，从数学的角度来看，潜在的方程可能是非常有趣和具有挑战性的，因为它们的解可以表现出丰富的行为，例如图案形成和行波结构。解的行为反映了系统的关键特征，这些特征是数学建模的，并与真实世界系统的演化具有明确和令人信服的相似之处。然而，现实世界的系统与其更易处理的数学模型之间仍有很大差距。该项目的目标之一是通过使用积累的具体数据以及对数学模型进行相应的校准和修改来弥合这一差距。尤其重要的是对包括异质环境的系统的分析，了解非局部扩散对解决方案行为的影响，以及用真实世界数据验证这些模型。将侧重于四个反应-平流-扩散系统，其异质性的原因是：生态背景下的气候变化、暴乱背景下的非本地和不对称的信息传播、社会隔离背景下的收入异质性以及特定集中投入造成的环境异质性。该项目的主要目标是了解异质环境和非局部扩散如何影响一般类型的非线性反应扩散方程以及在特定的生态和社会背景下的解模式。在整个项目中，研究人员将保持和促进与社会科学家的联系，以便在数学理论和应用之间进行亟需的讨论。该项目将致力于发展非均质环境中反应-平流-扩散系统的广泛理论，并将其应用于生态学和社会学。具体地说，这项研究的目标有三个：扩展非均匀环境中局部和非局部反应-平流-扩散系统的现有数学理论，通过使用这些类型的系统对各种(生态、社会学和生物)复杂系统进行建模来深入了解它们，并通过结合使用数据来弥合基本数学模型和它们旨在描述的复杂现实世界系统之间的差距。从定性的角度来看，这项工作将主要致力于探索各种扩散机制对非均质环境中解的传播(以及缺乏)的影响：例如，确定非均质系统中的传播速度、行波解、脉动前锋、行波脉冲或广义前锋的存在。此外，还将严格分析这些系统解的全局适定性、中长期渐近性以及非平凡稳态解的存在唯一性等基本问题。

This research is aimed at development of mathematical tools for understanding and predicting behavior of a variety of important and apparently dissimilar biological phenomena and sociological processes, such as the spread of diseases, ecological changes, or the distribution and prevention of crime. Some salient features of these diverse phenomena can be described by mathematical models based on reaction-advection-diffusion equations that have been used extensively to investigate fundamental and ubiquitous phenomena in several areas of biology, and more recently in the social sciences. While these models are simplified versions of reality, their mathematical analysis has contributed to the understanding of many important phenomena. At the same time, the underlying equations can be extremely interesting and challenging from the mathematical point of view as their solutions can exhibit rich behaviors, e.g., pattern formation and traveling wave structures. The behavior of solutions reflects critical features of the system that mathematics models and have clear and convincing parallels with the evolution of real-world systems. However, there is still a large gap between real world systems and their more tractable mathematical models. One of the goals of this project is bridging this gap through the use of accumulated concrete data and the corresponding calibration and modification of mathematical models. Of particular importance is the analysis of systems which include heterogeneous environments, the understanding of the effects that non-local dispersal has on the behavior of the solutions, and the validation of these models with real-world data. Efforts will be focused on four reaction-advection-diffusion systems where the heterogeneities are due to: climate change in an ecological context, non-local and asymmetrical spread of information in the context of riots, income heterogeneities in the context of social segregation, and environment heterogeneities due to specific concentrated inputs. The primary goal of the project is to understand how heterogeneous environments and non-local dispersal impact solution patterns in both a general class of nonlinear reaction-diffusion equations as well in specific ecological and social contexts. Throughout this project the investigators will maintain and foster contacts with social scientists in order to conduct the much needed discourse between the mathematical theory and the applications. This project will focus on the development of an extensive theory for reaction-advection-diffusion systems in heterogeneous environments with applications in ecology and sociology. In particular, the objective of this research is three-fold: to expand the current mathematical theory for local and non-local reaction-advection-diffusion systems in heterogeneous environments, to gain insight into various (ecological, sociological, and biological) complex systems by modeling them using these types of systems, and to take an initial step toward bridging the gap between basic mathematical models and the complex real-world systems they aim to describe by incorporating the use of data. From the qualitative perspective, this work will be mainly concerned with exploring the effects that various dispersal mechanism have on the propagation (and lack thereof) of a solution in a heterogeneous environment: for example, determining the spreading speed, the existence of traveling wave solutions, pulsating fronts, traveling pulses, or generalized fronts in heterogeneous systems. Additionally, the fundamental issues of the global well-posedness of solutions to these systems, intermediate and long-term asymptotics, and existence and uniqueness of non-trivial steady-state solutions will be rigorously analyzed.